10/19/2019
Colin Grudzien1,3, Marc Bocquet2 and Alberto Carrassi3,4
used to combine simulations from a physics-based numerical model and real-world observations of a physical process.
The output of DA is an estimate of a posterior probability density for the numerically simulated physical state, or some statistic of it.
In this Bayesian framework, an ensemble-based forecast represents a sampling procedure for the forecast-prior probability density.
The process of sequentially and recursively estimating the distribution for the system's state by combining model forecasts and streaming observations is known as filtering.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
From: Carrassi, A. et al. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdisciplinary Reviews: Climate Change 9.5 (2018): e535.
Due to the large dimensionality and complexity of operational geophysical models, an accurate representation of the true Bayesian posterior is infeasible.
Therefore, the DA cycles typically estimates the first two moments of the posterior, or its mode.
Due to the expense of generating model forecasts, the practical number of samples is typically on the order of \( \mathcal{O}\left(10^2\right) \).
A Gaussian approximation for the prior, posterior and likelihoods is at the basis of all operational DA schemes, due to the operational constraints.
In a linear-Gaussian model, the effect of estimating the mean or the mode would be the same;
Therefore, different filtering assumptions and approximations lead to dramatically different results in practice.
Toy models are small-scale numerical analogues of full-scale geophysical dynamics which are used in twin experiments.
Courtesy of: Kaidor via Wikimedia Commons (CC 3.0)
From: Wilks, D. Effects of stochastic parametrizations in the Lorenz'96 system. Quarterly Journal of the Royal Meteorological Society 131.606 (2005): 389-407.
The two layer Lorenz-96 model is commonly used in benchmark twin experiments thus to study the effects of model uncertainty and model reduction errors in multiscale dynamics.
It has recently been shown in a deterministic, biased-model setting that the numerical precision of the discretization of the ensemble forecast can be significantly reduced without a major deterioration of the DA cycle's (relative) predictive performance3.
However, important differences in the statistical properties of model forecasts of stochastic dynamical systems have been observed due to the discretization errors of certain low-order schemes.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
We confirm the overall robustness of the Runge-Kutta scheme in generating ensemble-forecast statistics in the L96-s model.
We note that even for a coarse time step of \( \Delta=10^{-2} \), the Runge-Kutta scheme has an unbiased spread compared with the Taylor scheme using a step size of \( \Delta=10^{-3} \).
On the other hand, the Euler-Maruyama scheme introduces systematic biases into ensemble forecast statistics.
This is concerning as the Euler-Maruyama scheme is commonly used to simulate systems of SDEs for twin experiments.
While this indicates the biasing of ensemble forecast statistics, this doesn't yet demonstrate the effect of this bias on a filter twin experiment.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
From: Grudzien et al. On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments. In submission.
Typically, the classical Lorenz-96 system is simulated with a 4-stage Runge-Kutta scheme with step size up to \( 0.5 \) in deterministic settings.
Particularly, we must make a distinction between strong and weak convergence, and its impact on the truth-twin and the model-twin-respectively.
While Euler-Maruyama is a commonly used integration scheme for SDEs, we find that it introduces strong, systematic bias into twin experiments when the step size is greater than or equal to \( \Delta=10^{-2} \).
On the other hand, we find that the 4-stage Runge-Kutta scheme is a statistically robust solver, without the systematic biases encountered in the Euler-Maruyama scheme.